The Convergence Of Wavelet Series

In this paper we discuss the convergence of wavelet expansions. The paper includes two parts. In the first part we discuss the convergence of wavelet expansions in frequency space. In the second part we discuss the convergence of convolution operators of Shannon type.In her thesis of master degree Tang Manhui proved the following result:suppose that f∈L2∩L1and Φ is a scaling function with Φ∈RB. Then the Fourier transform Pmf(ω) of multi-resolution expansion of f is convergent to f(ω) almost everywhere. We consider the case where Φ∈RB does not hold. Namely we proved the following three results.1. Assume that Φ∈L2is a scaling function with|Φ|2∈RB. If f∈L2∩L1satisfies|f|2∈RB, then the Fourier transform Pmj(ω) of multi-resolution expansion of f is convergent to f(ω) almost everywhere.2. Assume that Φ∈L2is a scaling function with|Φ|2∈RB. If f∈L2∩L1satisfies∑j∈z|f(x+j)|∈L2(0,1), then the Fourier transform Pmf(ω) of multi-resolution expansion of f is convergent to f(ω) almost everywhere.3. Assume that Φ∈L2is a scaling function with|Φ|2∈RB and satisfies|Φ(ξ)|≤c/((1+|ξ|1-α)(0<α<1). If f∈L2∩L1and its modulos of continuity in L1ω1(f,t)=O(t3)(tâ†'0+),where β>α. then the Fourier transform Pmf|(ω) of multi-resolution expansion of f is convergent to f(ω) almost everywhere.